An extension of the angular synchronization problem to the heterogeneous setting

Given an undirected measurement graph G = ([n], E), the classical angular synchronization problem consists of recovering unknown angles θ1,dots,θn from a collection of noisy pairwise measurements of the form (θi - θj) mod 2π, for each i,j ∈ E. This problem arises in a variety of applications, including computer vision, time synchronization of distributed networks, and ranking from preference relationships. In this paper, we consider a generalization to the setting where there exist k unknown groups of angles θl,1, dots,θl,n, for l=1,dots,k. For each i,j ∈ E, we are given noisy pairwise measurements of the form θℓ,i - θℓ,j for an unknown ℓ ∈ 1,2,ldots,k. This can be thought of as a natural extension of the angular synchronization problem to the heterogeneous setting of multiple groups of angles, where the measurement graph has an unknown edge-disjoint decomposition G = G1 cup G2 ldots cup Gk, where the Gi's denote the subgraphs of edges corresponding to each group. We propose a probabilistic generative model for this problem, along with a spectral algorithm for which we provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise. The theoretical findings are complemented by a comprehensive set of numerical experiments, showcasing the efficacy of our algorithm under various parameter regimes. Finally, we consider an application of bi-synchronization to the graph realization problem, and provide along the way an iterative graph disentangling procedure that uncovers the subgraphs Gi, i=1,ldots,k which is of independent interest, as it is shown to improve the final recovery accuracy across all the experiments considered.